High-dimensional modeling via nonconcave penalized likelihood and local likelihood

This dissertation consists of two parts: model selection via nonconcave penalized likelihood and statistical modeling via local likelihood. In Part I of this dissertation, a few variable selection procedures via penalized likelihood are proposed. The proposed methods select variables and estimate coefficients simultaneously. Hence it is easy to construct confidence intervals for estimated parameters. A new algorithm is proposed for optimizing high-dimensional nonconcave functions. The proposed ideas are widely applicable. They are readily applied to a variety of parametric models and semi-parametric models. They can also be applied to nonparametric modeling by using wavelets and splines. It has been demonstrated how the rates of convergence for the proposed estimators depend on the regularization parameter. Oracle properties have been established for the proposed variable selection procedures in this dissertation. In Part II of this dissertation, an efficient estimation procedure is proposed to estimate coefficient functions in varying-coefficient models. The asymptotic normality of the resulting estimators is established. The standard error formulae for estimated coefficients are derived and empirically tested. A goodness-of-fit test technique, based on a nonparametric maximum likelihood ratio type of test, is also proposed to detect whether certain coefficient functions in a generalized varying-coefficient model are constant or whether any covariates are statistically significant in the model. The null distribution of the test is estimated by a conditional bootstrap method. To reduce computational burden, a one-step Newton-Raphson estimator is proposed and implemented. It is shown that the resulting one-step procedure can save computational cost by a factor of tens without deteriorating its performance, both asymptotically and empirically. The SiZer map has been developed by Chaudhuri and Marron (1999) as a methodology for finding which features in noisy data are strong enough to be distinguished from background noise. An extension of the SiZer map with more efficiency in distinguishing features is proposed in Chapter 7.